We end this section by noting that in the graph signal processing literature [1]}, a controller of the form of (REF ) is known as a linear, shift-invariant (LSI) graph filter, and is analogous to an LTI filter.
Note that \(S \mathbf {\Phi }^\star \mathbf {x}= \mathbf {\Phi }^\star S \mathbf {x}\) , hence the name.
In particular, Equation (REF ) is a spatially finite impulse response (FIR) graph filter [2]} that is completely characterized by a finite set of \(N\)  filter taps that can be conveniently described by a collection of transfer functions \(\phi ^{\star } = [\phi _{0}^{\star },\ldots ,\phi _{N-1}^{\star }]^{\mathsf {T}} \in \mathbb {R}^{N}\) . We emphasize that the transfer functions themselves are not restricted to be temporally FIR.
Spatially FIR graph filters are also known as convolutional graph filters [3]} due to their sum-and-shift nature, understanding that the effect of the operation \(S \mathbf {x}\)  is to shift the signal around the graph (thus, oftentimes, the GMD \(S\)  is referred to as the graph shift operator).
Furthermore, spatially FIR graph filters satisfy the convolution theorem that indicates that a convolution in the vertex domain can be computed by means of an elementwise multiplication in the spectrum domain [1]}.
Finally, in the context of finite graphs, it is observed that the space of FIR graph filters of the form (REF ), characterized by \(N\)  filter taps, is equivalent to the space of spatially infinite impulse response (IIR) graph filters as well as autoregressive, moving average (ARMA) graph filters [5]}.
